Finite Group Databasebeta

← C3×#27756 #10646363

Subgroups — C3×#27756 #10646363

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2552C211yesmaximal, characteristic, complemented
2C3×#2547C211yesmaximal, characteristic
3C3×#2072C212yesmaximal, complemented
4C3×#2042C212yesmaximal
5C3×#1981C211yesmaximal, characteristic, complemented
6C3×#1959C212yesmaximal
7C3×#1956C211yesmaximal, characteristic, complemented
8C3×#1935C212yesmaximal, complemented
9C3×#1802C211yesmaximal, characteristic
10C3×#1770C211yesmaximal, characteristic, complemented
11C3×C4×#102C211yesmaximal, characteristic
12#27756C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125—21no—
15C2×C3×#124C2212yes—
16C2×C3×C4×D4C2211yescharacteristic
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#103C2212yes—
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2212yes—
25C2×C3×#97C2212yes—
26C2×C3×C4×C8C2211yescharacteristic
27C3×#398—22no—
28C3×#397—22no—
29C3×#391C2212yes—
30C3×#389—21no—
31C3×#389C2212yes—
32C3×#384C2211yescharacteristic
33C3×#384C2212yes—
34C3×#380—21no—
35C3×#380—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×#340C2211yescharacteristic
39C3×#340C2211yescharacteristic
40C3×#340C2212yes—
41C3×#336C2212yes—
42#2552C2×C311yescharacteristic, complemented
43#2547C2×C311yescharacteristic
44#2072C2×C312yescomplemented
45#2042C2×C312yes—
46#1981C2×C311yescharacteristic, complemented
47#1959C2×C312yes—
48#1956C2×C311yescharacteristic, complemented
49#1935C2×C312yescomplemented
50#1802C2×C311yescharacteristic
51#1770C2×C311yescharacteristic, complemented
52C4×#102C2×C311yescharacteristic
53C22×C3×Q8C2312yes—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2312yes—
58C22×C3×C8C2311yescharacteristic
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×#125—44no—
62C3×#124—24no—
63C3×#124—42no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D411yescharacteristic
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—21no—
83C2×C3×C42C2312yes—
84C2×C3×C42C2311yescharacteristic
85C3×C4⋊C8—22no—
86C3×#103—24no—
87C3×#103—42no—
88C3×#103—42no—
89C3×#103—42no—
90C3×#102—22no—
91C3×#102—22no—
92C3×#102—42no—
93C3×#97—42no—
94C3×C4×C8—22no—
95C3×#95—21no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95D412yes—
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×#125—21no—
102C2×#124C22×C312yes—
103C2×C4×D4C22×C311yescharacteristic
104C2×C4⋊C8C22×C311yescharacteristic
105C2×#103C22×C312yes—
106C2×#103C22×C312yes—
107C2×#103C22×C312yes—
108C2×#103C22×C312yes—
109C2×#102C22×C311yescharacteristic
110C2×#102C22×C311yescharacteristic
111C2×#102C22×C312yes—
112C2×#97C22×C312yes—
113C2×C4×C8C22×C311yescharacteristic
114#398—22no—
115#397—22no—
116#391C22×C312yes—
117#389—21no—
118#389C22×C312yes—
119#384C22×C311yescharacteristic
120#384C22×C312yes—
121#380—21no—
122#380—21no—
123C4×C4⋊C4C22×C311yescharacteristic
124C4×C4⋊C4C22×C311yescharacteristic
125#340C22×C312yes—
126#340C22×C311yescharacteristic
127#340C22×C311yescharacteristic
128#336C22×C312yes—
129C24×C3—21no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4#4112yes—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—21no—
163C2×C3×C8#4112yes—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—81no—
180C3×C4⋊C4—82no—
181C3×C22⋊C4—44no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—42no—
184C3×C42—22no—
185C3×C42—22no—
186C3×C42—42no—
187C3×C42#4112yes—
188C3×C42#4112yes—
189C22×Q8C23×C312yes—
190C22×D4C23×C311yescharacteristic
191C23×C4—21no—
192C22×C8C23×C311yescharacteristic
193C22×C8C23×C312yes—
194C22×C8C23×C311yescharacteristic
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197#125—44no—
198#124—24no—
199#124—42no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C3×D411yescharacteristic
210C2×C4⋊C4C3×D411yescharacteristic
211C2×C4⋊C4C23×C311yescharacteristic
212C2×C4⋊C4C23×C311yescharacteristic
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C4⋊C4C23×C312yes—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—21no—
219C2×C42C23×C312yes—
220C2×C42C23×C311yescharacteristic
221C4⋊C8—22no—
222#103—24no—
223#103—42no—
224#103—42no—
225#103—42no—
226#102—22no—
227#102—22no—
228#102—42no—
229#97—42no—
230C4×C8—22no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95C3×D412yes—
236C23×C3—22no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3C22×D411yescharacteristic
244C3×Q8—42no—
245C3×Q8—42no—
246C3×Q8—42no—
247C3×Q8—42no—
248C3×D4—22no—
249C3×D4—22no—
250C3×D4—41no—
251C3×D4—41no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—84no—
276C2×C3×C4#12412yes—
277C2×C3×C4#12412yes—
278C2×C3×C4AGL(1,ℤ/8)12yes—
279C2×C3×C4#13712yes—
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4H(5,𝔽2)11yescharacteristic
285C2×C3×C4H(5,𝔽2)11yescharacteristic
286C3×C8—22no—
287C3×C8—42no—
288C3×C8—41no—
289C24—21no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4C3×#4112yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4C2×C3×D412yes—
311C22×C4C2×C3×D412yes—
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C24×C311yescharacteristic
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8C3×#4112yes—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—81no—
340C4⋊C4—82no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—42no—
344C42—22no—
345C42—22no—
346C42—42no—
347C42C3×#4112yes—
348C42C3×#4112yes—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3C2×#12411yescharacteristic
357C22×C3#54611yescharacteristic
358C22×C3#55011yescharacteristic
359C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C3C2×#13711yescharacteristic
361C22×C3#57711yescharacteristic
362C22×C3#57811yescharacteristic
363C3×C4—22no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—81no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4—82no—
370C3×C4—82no—
371C3×C4#48914yes—
372C3×C4#52911yescharacteristic
373C3×C4#54011yescharacteristic
374C3×C4#55311yescharacteristic
375C3×C4#55311yescharacteristic
376C23—22no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23C22×C3×D411yescharacteristic
384Q8—42no—
385Q8—42no—
386Q8—42no—
387Q8—42no—
388D4—22no—
389D4—22no—
390D4—41no—
391D4—41no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—84no—
416C2×C4C3×#12412yes—
417C2×C4C3×#12412yes—
418C2×C4C3×AGL(1,ℤ/8)12yes—
419C2×C4C3×#13712yes—
420C2×C4C2×C3×#4111yescharacteristic
421C2×C4C2×C3×#4111yescharacteristic
422C2×C4C2×C3×#4111yescharacteristic
423C2×C4C2×C3×#4111yescharacteristic
424C2×C4C3×H(5,𝔽2)11yescharacteristic
425C2×C4C3×H(5,𝔽2)11yescharacteristic
426C8—22no—
427C8—42no—
428C8—41no—
429C2×C3—82no—
430C2×C3#249011yescharacteristic
431C2×C3C2×#48911yescharacteristic
432C2×C3#313711yescharacteristic
433C2×C3#333011yescharacteristic
434C2×C3#333111yescharacteristic
435C2×C3#339711yescharacteristic
436C2×C3#339811yescharacteristic
437C22—42no—
438C22—42no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22C2×C3×#12411yescharacteristic
445C22C3×#54611yescharacteristic
446C22C3×#55011yescharacteristic
447C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
448C22C2×C3×#13711yescharacteristic
449C22C3×#57711yescharacteristic
450C22C3×#57811yescharacteristic
451C4—22no—
452C4—42no—
453C4—42no—
454C4—81no—
455C4—81no—
456C4—82no—
457C4—82no—
458C4—81no—
459C4C3×#48914yes—
460C4C3×#52911yescharacteristic
461C4C3×#54011yescharacteristic
462C4C3×#55311yescharacteristic
463C4C3×#55311yescharacteristic
464C3#2775611yescharacteristic, direct product factor, Sylow
465C2—82no—
466C2C3×#249011yescharacteristic
467C2C2×C3×#48911yescharacteristic
468C2C3×#313711yescharacteristic
469C2C3×#333011yescharacteristic
470C2C3×#333111yescharacteristic
471C2C3×#339711yescharacteristic
472C2C3×#339811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#2552C2semidirect
3. C3×#2072C2semidirect
5. C3×#1981C2semidirect
7. C3×#1956C2semidirect
8. C3×#1935C2semidirect
10. C3×#1770C2semidirect
12. #27756C3direct (N × H)
42. #2552C2×C3semidirect
44. #2072C2×C3semidirect
46. #1981C2×C3semidirect
48. #1956C2×C3semidirect
49. #1935C2×C3semidirect
51. #1770C2×C3semidirect
464. C3#27756direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#25522
C3×#25472
C3×#20722
C3×#20422
C3×#19812
C3×#19592
C3×#19562
C3×#19352
C3×#18022
C3×#17702
C3×C4×#1022
#277563